• Black Bifurcation physics hoodie, pixel-art logistic map period doubling design, worn by a model, front close-up
  • Black Bifurcation physics hoodie, pixel-art logistic map period doubling design, worn by a model, back close-up
  • Black Bifurcation physics hoodie, pixel-art logistic map period doubling design, front view
  • Black Bifurcation physics hoodie, pixel-art logistic map period doubling design, back view
  • Black Bifurcation physics hoodie, pixel-art logistic map period doubling design, turning on a model
  • Black Bifurcation physics hoodie, ΨP Pixelated Physics neck label
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Bifurcation HoodieSPLIT, SPLIT, SPLIT, CHAOS | Pixelated Physics

Bifurcation Hoodie – SPLIT, SPLIT, SPLIT, CHAOS | Pixelated Physics

$54.99 USD
$54.99 USD
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Shipping calculated at checkout. Free US shipping · $10 flat shipping outside the US.
Color
Black
Size
S–5XL
Made
To order · shipping times

One line of arithmetic: multiply by r, multiply by one minus itself, repeat. The logistic map x → r x(1 − x) from r = 2.8 to 4: period doubling at a rate set by δ = 4.669…, then chaos with windows of order. Pixel-art unisex hoodie in black; equation, symbols and sources in the physics panel below.

Color
Size

Size guideS–5XL · inches

Details

Size & fitS–5XL

Unisex Heavy Blend hoodie (Gildan 18500), classic fit.

SizeWidthLengthSleeve
S202733.5
M222834.5
L242935.5
XL263036.5
2XL283137.5
3XL303238.5
4XL323339.5
5XL343440.5

Measurements in inches. Width and length ±1 in, sleeve (from center back) ±0.75 in.

ShippingFree US

Free US shipping. $10 flat shipping outside the US.

Delivery times & where we ship

Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

One line of arithmetic: multiply by r, multiply by one minus itself, repeat.

For small r the value settles on a single number. At r = 3 it starts alternating between two, then four, then eight. Each split comes sooner than the last, by a factor that tends to δ = 4.669…, the same number for a whole class of maps. Mitchell Feigenbaum found that in 1978.

Past r ≈ 3.5699 the orbit never settles. The art plots where it lands after a long transient, for each r, shaded by how often. Inside the chaos are thin windows where order returns, the widest a period-3 window at r = 1 + √8.

Robert May brought the map to ecologists in 1976 as a warning: very simple models can do very complicated things.

Split, split, split, chaos.

Equations

\[x_{n+1} = r\,x_n(1-x_n)\]
\[\delta = \lim_{k\to\infty}\frac{r_k - r_{k-1}}{r_{k+1}-r_k} = 4.6692\ldots\]
SymbolMeaningUnit
\(x_n\) population as a fraction of its maximum, step n \(\mathrm{1}\)
\(r\) growth parameter \(\mathrm{1}\)
\(r_k\) value of r at the k-th period doubling \(\mathrm{1}\)
\(\delta\) Feigenbaum constant \(\mathrm{1}\)

Sources

  1. May (1976) Simple mathematical models with very complicated dynamics, Nature 261, 459
  2. Feigenbaum (1978) Quantitative universality for a class of nonlinear transformations, J. Stat. Phys. 19, 25
  3. Strogatz, Nonlinear Dynamics and Chaos, Ch. 10
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